References
- B. Al-Hdaibat, W. Govaerts, Yu.A. Kuznetsov, and H.G.E. Meijer, Initialization of homoclinic solutions near Bogdanov--Takens points: Lindstedt--Poincaré compared with regular perturbation method, SIAM J. Appl. Dyn. Syst. 15 (2016), pp. 952–980.
- V.I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations, Springer-Verlag, London, 1983.
- D.K. Arrowsmith, J.H.E. Cartwright, A.N. Lansbury, and C.M. Place, The Bogdanov map: Bifurcations, mode locking, and chaos in a dissipative system, Int. J. Bifurcation Chaos 03 (1993), pp. 803–842.
- W.-J. Beyn, Numerical analysis of homoclinic orbits emanating from a Takens-Bogdanov point, IMA J. Numer. Anal. 14 (1994), pp. 381–410.
- W.-J. Beyn, T. Hüls, and Y. Zou, Numerical analysis of degenerate connecting orbits for maps, Int. J. Bifurcation Chaos 14 (2004), pp. 3385–3407.
- W.-J. Beyn and J.-M. Kleinkauf, The numerical computation of homoclinic orbits for maps, SIAM J. Numer. Anal. 34 (1996), pp. 1207–1236.
- H.Broer, R.Roussarie, and C.Simó, Invariant circles in the Bogdanov–Takens bifurcation for diffeomorphisms, Ergodic Theor. Dyn. Syst. 16 (1996), pp. 1147–1172.
- J. England, B. Krauskopf, and H. Osinga, Computing one-dimensional stable manifolds and stable sets of planar maps without the inverse, SIAM J. Appl. Dyn. Syst. 3 (2004), pp. 161–190.
- V. Gelfreich, Chaotic zone in the Bogdanov-Takens bifurcation for diffeomorphisms, in Analysis and Applications - ISAAC 2001, Vol. 10, International Society for Analysis, Applications and Computation, p, H.G.W. Begehr, R.P. Gilbert and M.W. Wong, eds., Springer, New York, 2003, pp. 187–197.
- V. Gelfreich and V. Naudot, Analytic invariants associated with a parabolic fixed point in x∈ℂ2, Ergodic Theor. Dyn. Syst.. 28 (2008), pp. 1815–1848.
- V. Gelfreich and V. Naudot, Width of homoclinic zone for quadratic maps, Exp. Math. 18 (2009), pp. 409–427.
- M.L. Glasser, V.G. Papageorgiou, and T.C. Bountis, Mel’nikov’s function for two-dimensional mappings, SIAM J. Appl. Math. 49 (1989), pp. 692–703.
- J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer-Verlag, New York, 1983.
- R. Khoshsiar Ghaziani, Bifurcations of maps: Numerical algorithms and applications, Ph.D. thesis., Ghent University, 2008.
- R. Khoshsiar Ghaziani, W. Govaerts, Yu. A. Kuznetsov, and H.G.E. Meijer, Numerical continuation of connecting orbits of maps in MATLAB, J. Differ. Equ. Appl. 15 (2009), pp. 849–875.
- Yu.A. Kuznetsov, Elements of Applied Bifurcation Theory, 3rd ed., Springer-Verlag, New York, 2004.
- Yu.A. Kuznetsov, H.G.E. Meijer, B. Al-Hdaibat, and W. Govaerts, Accurate approximation of homoclinic solutions in Gray-Scott kinetic model, Int. J. Bifurcation Chaos. 25 (2015), Article No. 1550125, 10p.
- Ju.I. Neĭmark, The Method of Point Transformations in the Theory of Nonlinear Oscillations. Nauka, Moscow, 1972. [Russian].
- J.N. Páez, Chávez. Starting homoclinic tangencies near 1:1 resonances, Int. J. Bifurcation Chaos 20 (2010), pp. 3157–3172.